#### The value of $\sum_{\mathrm{r}=0}^{2 \mathrm{n}}(-1)^{\mathrm{r}} \cdot\left({ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{r}}\right)^2$ is equal toOption: 1 ${ }^{4 \mathrm{n}} \mathrm{C}_{2 \mathrm{n}}$Option: 2 $(-1)^{\mathrm{n}} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}$Option: 3 ${ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{a}}$Option: 4 none of these

\begin{aligned} & \sum_{\mathrm{r}=0}^{2 \mathrm{n}}(-1)^{\mathrm{r}} \cdot\left({ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{r}}\right)^2=\sum_{\mathrm{r}=0}^{2 \mathrm{n}}(-1)^{\mathrm{r}} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{r}} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{r}} \\ & =\sum_{\mathrm{r}=0}^{2 \mathrm{n}}(-1)^{\mathrm{r}} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{r}} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{2 \mathrm{n}-\mathrm{r}} \\ & =\text { coefficient of } \mathrm{x}^{2 \mathrm{n}} \text { in }(1-\mathrm{x})^{2 \mathrm{n}} \cdot(1+\mathrm{x})^{2 \mathrm{n}} \\ & =\text { coefficient of } \mathrm{x}^{2 \mathrm{n}} \text { in }\left(1-\mathrm{x}^2\right)^{2 \mathrm{n}}=(-1)^{\mathrm{n}} \cdot{ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}} \cdot \end{aligned}